INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 16 (2005) S118–S124
doi:10.1088/0957-4484/16/3/022
Investigating atomic details of the
CaF2(111) surface with a qPlus sensor
Franz J Giessibl1 and Michael Reichling2
1
Universität Augsburg, Experimentalphysik 6, EKM, Institut für Physik, 86135 Augsburg,
Germany
2
Fachbereich Physik, Universität Osnabrück, Barbarastraße 7, 49076 Osnabrück, Germany
Received 18 November 2004, in final form 15 December 2004
Published 28 January 2005
Online at stacks.iop.org/Nano/16/S118
Abstract
The (111) surface of CaF2 has been intensively studied with large-amplitude
frequency-modulation atomic force microscopy, and the atomic contrast
formation is now well understood. It has been shown that the apparent
contrast patterns obtained with a polar tip strongly depend on the tip
terminating ion, and three sub-lattices of anions and cations can be imaged.
Here, we study the details of atomic contrast formation on CaF2 (111) with
small-amplitude force microscopy utilizing the qPlus sensor that has been
shown to provide the utmost resolution at high scanning stability. Step edges
resulting from cleaving crystals in situ in the ultra-high vacuum appear as
very sharp structures, and on flat terraces the atomic corrugation is seen in
high clarity even for large area scans. The atomic structure is also not lost
when scanning across triple layer step edges. High-resolution scans of small
surface areas yield contrast features of anion- and cation sub-lattices with
unprecedented resolution. These contrast patterns are related to previously
reported theoretical results.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
CaF2 is an important material for science and technology, for
example as a lens material for 157 nm lithography [1] or as
a high-bandgap insulating layer with an almost perfect lattice
match for epitaxial insulating layers on silicon [2]. A detailed
knowledge of its surface structure and defects is important.
While thin CaF2 layers can be imaged by scanning tunnelling
microscopy [3], an atomic force microscope (AFM) [4] is
required for imaging thicker layers [5] or bulk materials [6].
Crystalline CaF2 has a face-centred cubic lattice (see figure 1).
The natural cleavage planes are the {111} planes [7]; the
−
corresponding surface
√ layers are trigonal arrangements of F ions spaced by a0 / 2 = 386.2 pm. Electrostatic energy
considerations lead to the conclusion that the CaF2 (111)
surface must be terminated by a complete triple layer
F− –Ca2+ –F− with a F− -layer at the surface [7]. In contrast
to the (001) cleavage planes of alkali halides, the CaF2 (111)
surface offers a reference sample where the atomic contrast in
experimental images is tightly connected to the signature of the
electric charge on the tip [8]. AFM studies of the CaF2 surface
are available on the atomic scale [6, 9, 10] as well as on a larger
0957-4484/05/030118+07$30.00 © 2005 IOP Publishing Ltd
scale [11]. While a consistent understanding of the atomic
contrast has been achieved, step structures have so far proven
hard to be imaged with a traditional large-amplitude AFM at
atomic resolution, and theoretical considerations that predict
a change in contrast pattern when imaging in the repulsive
regime [8, 12] have so far not been verified experimentally.
The use of small amplitudes helps to attenuate the disturbing
long-range interaction forces [13] and enables non-destructive
atomic imaging in the repulsive regime [14]. The qPlus
sensor can be operated with high stability at sub-nanometre
amplitudes [15], motivating us to revisit CaF2 (111).
2. Challenge of long-range forces on steps
In an ideal (hypothetical) AFM, the probe would consist of a
single atom. In reality, the front atom needs to be supported
by other atoms, forming a mesoscopic tip that is connected
to a cantilever. The tip–sample forces in an AFM are a sum
of long-range and short-range contributions. While chemical
bonding forces between single atoms decay exponentially with
increasing distance, van-der-Waals (vdW) and electrostatic
forces decay following an inverse power law, and therefore
Printed in the UK
S118
Investigating atomic details of the CaF2 (111) surface with a qPlus sensor
between the tip and the sample is typically significantly greater
than the chemical bonding force between the front atom and
the sample. The long-range contribution is often modelled by
a vdW interaction of a spherical tip with radius R, yielding a
long-range force given by [16]:
FvdW (z) = −
AH R
6z 2
(1)
where AH is the Hamaker constant. In frequency modulation
AFM [17], the tip–sample force is not measured directly.
Instead, the averaged gradient of the tip–sample force with
respect to the surface normal leads to a frequency shift f of
a cantilever with an unperturbed eigenfrequency f0 , spring
constant k and oscillation amplitude A. A vdW force as
in equation (1) leads to a normalized frequency shift γ =
( f / f0 )k A3/2 given by [18]:
γvdW (z) = − AH R
Figure 1. Crystal structure of CaF2 , forming a face-centred cubic
lattice with a lattice constant of a0 = 546 pm. The Ca 2+ -ions are
represented by green spheres, the F− -ions by blue spheres. The
basis of the lattice consists of three atoms with a Ca 2+ -ion located at
the origin of the fcc lattice and two F− -ions located at
(x,
√ y, z) = ±(a0 /4, a0 /4, a0 /4) with an interionic distance of
3/4 a0 = 237 pm. (a) Perspective view of the cubic unit cell.
(b) Top view of a CaF2 (111) surface with a surface layer consisting
−
of F− -ions. The
√ surface layer is a trigonal arrangement of F -ions
spaced by a0 / 2 = 386 pm; the angle between the surface lattice
vectors is 60◦ . (c) View parallel to the surface along the [1̄10]
direction, showing the electrically
√ neutral triple layers of
F− –Ca 2+ –F− ions spaced by a0 / 3 = 315 pm. Because the surface
should be terminated by complete triple layers (see the text), the step
heights should be integer multiples of 315 pm. The Ca + -layer is
79 pm below the surface F− -layer, and the second F− -layer is
158 pm lower than the surface layer.
have a much longer range. While chemical bonding forces
for two atoms at a close distance can be much greater than
the vdW forces between the two atoms, the total vdW force
A3/2
.
6(z 2 + 2Az)3/2
(2)
The long-range force is proportional to the tip radius R,
and thus sharp tips minimize the long-range contribution
to the normalized frequency shift. This long-range force
is a challenge for high-resolution AFM, in particular when
scanning across steps where the long-range force changes
as a function of the lateral sample position. Guggisberg
et al [19] have investigated the frequency shift difference on
upper and lower terraces (FREDUL) in Si(111) and found a
voltage dependent variation corresponding to 5 fN m0.5 when
compensating the contact potential difference and 30 fN m0.5
for a bias voltage of 2 V. As short-range contributions to γ are
of the order of 1 fN m0.5 , this clearly stresses the challenges
faced when attempting AFM at atomic resolution across steps.
While sporadic reports of atomic resolution across step edges
by large amplitude FM-AFM have been reported (e.g. [20, 21]),
it requires a cantilever with an exceptionally sharp tip. Here
we show that when using small amplitudes, atomic resolution
across step edges is even possible with relatively blunt tips.
In the beginning of an AFM experiment, the tip often
collects a cluster consisting of sample material with a height ,
which also reduces the long-range force. Figure 2(a) shows the
calculated dependence of the long-range component of γ as a
function of cluster height and amplitude A after equation (2).
It is clearly evident that the long-range contribution of γ is
greatly reduced when using small amplitudes and having a
high tip cluster.
For short-range forces caused by covalent bonding, a
Morse potential
Vts (z) = E bond (−2 exp−κ(z−σ ) + exp−2κ(z−σ ) )
(3)
is a fair approximation for the short-range part of the tip–
sample interaction [22], where E bond is the bonding energy,
κ is the inverse interaction range and σ is the equilibrium
distance. When imaging ionic crystals, the dominant shortrange force is electrostatic in origin [12], but due to the
periodic arrangement of positive and negative charges, the net
electrostatic force decays roughly exponentially with a decay
constant of κ = 2π/aS [23], where aS is the length of the
surface unit vector. If the tip atom is not a single charged
atom, but a cluster of the sample material, the decay rate of the
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F J Giessibl and M Reichling
Figure 3. Schematic true-to-scale representation of the lower
section of a spherical tip with a radius of 100 nm and a
micro-asperity with a height of 2 nm close to a sample with a
315 pm step.
Figure 2. (a) Simulated normalized frequency shift γ as a function
of tip cluster height and amplitude A. (b) Simulated normalized
frequency shift γ and |γ | as a function of distance z for an
amplitude of A = 1 nm and the tip–sample forces described in
equations (1) and (3) with AH = 1 eV, R = 100 nm, = 1 nm,
F0 = 12.15 nN, κ = 16.3 nm−1 . (c) Logarithmic display of |γ |,
showing that a logarithmic filter provides a feedback signal that is
roughly linear with z for z-values outside of ≈0.21 nm.
electrostatic force is estimated at κ = 4π/aS . The short-range
part of the normalized frequency shift is given by
√
γsr (z) = F0 1/κ(−2 exp−κ(z−σ ) + 2 exp−2κ(z−σ ) ) (4)
where F0 is the tip–sample force at z = σ for amplitudes
larger than 1/κ [24]. It is noted that this model for the
short-range force is only qualitative, but it serves well to
discuss the challenges of atomic imaging. A detailed study
of the short-range forces in an AFM on CaF2 can be found
in references [8, 12]. Figure 2(b) shows the dependence of
the normalized frequency shift γ (z) for a tip–sample force
composed of long- and short-range contributions as described
by equations (1) and (3).
Rather than using f directly as a feedback signal, in our
experiment we routed f through a rectifier and a logarithmic
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filter before using it for feedback. Rectifying f prevents
catastrophic tip crashes due to inadvertent jumps into the
repulsive imaging regime as previously described by Ueyama
et al [25]. The use of a logarithmic filter improves tracking on
steps and other sharply inclined topography features, because
it provides a feedback signal that is more linear with distance
(see figure 2(c)). A disadvantage of rectifying f is that for
small magnitudes of γ (|γ | < 55 fN m0.5 in figure 2(c)),
a one-to-one relation between γ and z is not present, and
two z-values with ∂γ /∂z < 0 exist; thus the z-feedback
can find two z-values z i where the stable-feedback-conditions
γ (z i ) = γsetpoint and ∂γ /∂z < 0 for a distance regime
z = z i + are met. Figures 2(b) and (c) show that for
a setpoint of |γ | = 20 fN m0.5 , two distances are possible
(z 1 ≈ 0.2 nm and z 2 ≈ 0.4 nm). The z-intervals [z 1 − ,
z 1 + ] and [z 2 − , z 2 + ] where stable operation is possible
can have a width 2 reaching a few hundred picometres. In
some experiments we experienced jumps in topographic data
where stable topographic imaging was possible for two zvalues separated by approximately 0.2 nm. A switch from z 2 to
z 1 can be triggered intentionally when scanning rapidly across
a rising step, while a reverse switch is triggered by scanning
across a falling step (not shown here).
Figure 3 is a schematic view of a realistic tip over a step
edge. The attractive interaction is greater over the lower terrace
than over the higher terrace. This causes a challenge when
attempting to image step edges with atomic resolution. As
noted above, a reduction of the disturbing long-range force
can be achieved by using small oscillation amplitudes, high tip
clusters and sharp tips.
3. Experimental details
The sample used in this study was a CaF2 crystal with a size of
approximately 2 × 5 × 10 mm3 (Karl Korth, Kiel, Germany),
glued onto a 11 × 14 mm2 large sample holder plate. The
sample was cleaved along a predetermined breaking line in situ
in the (111) plane at a pressure of 5 × 10−7 Pa and transferred
within one minute to the microscope where it was kept at a
pressure of 5 × 10−8 Pa. As CaF2 (111) is not very reactive,
we obtained atomic resolution on this single cleave until five
days after cleaving, and collected approximately 4000 images
within that period.
The microscope (AutoProbe VP by Park Scientific
Instruments, Sunnyvale, USA) [26] was modified for qPlus
sensor operation. The force sensor is a standard qPlus
sensor [27] with a base frequency of f 0 = 16 740 Hz,
a Q-factor of 1700 and a stiffness of k = 1800 N m−1 .
An etched tungsten tip with an estimated tip radius of
100 nm was used as a tip. Frequency-to-voltage conversion
Investigating atomic details of the CaF2 (111) surface with a qPlus sensor
Figure 4. Overview AFM image and magnified views of a CaF2 (111) surface. Imaging parameters: scanning speed 96 nm s−1 ,
A = 625 pm, f = +7.32 Hz, γ = +12.3 fN m0.5 . (a) Image size 160 × 160 nm2 , with four different terrace heights spaced by integer
multiples (1–4) of 315 pm. (b) Image size 28.85 × 28.85 nm2 , showing a screw dislocation with a height of 315 pm. (c) Zoom into the step
at the screw dislocations; image size 8 × 8 nm2 showing atomic contrast. (d) Same image as (a) with five-fold increased contrast, showing
patched and linear structures on single terraces.
was done with a commercial phase-locked-loop detector
(EasyPLL by Nanosurf AG, Liestal, Switzerland). Images
were recorded at room temperature in the topographic
mode at constant frequency shift, supplemented by constantheight measurements of small sample sections that allow
us to estimate the long- and short-range contributions.
Drift correction of the acquired data was performed using
a commercial software package (SPIP Scanning Probe
Image Processor Version 2.21 by Image Metrology, Lyngby,
Denmark).
4. Results
We started the scan by slowly decreasing the setpoint of
f while monitoring the contrast. When performing AFM
measurements on CaF2 with soft cantilevers (k ≈ 10 N m−1 ),
surface charges usually cause unstable feedback conditions
and often the crystal is heated for a few hours to remove
these charges. Surprisingly, we did not have to heat the
sample to obtain stable imaging. Whether this is caused by
the large stiffness of the qPlus sensor or the metallic tip is
not yet determined. The large area scan shown in figure 4
shows flat terraces, separated by steps with heights of integer
multiples (1–4) of 320 ± 15 pm, in excellent agreement with
the expected triple-layer height of 315 pm. The data shown
in figure 4 are taken from a single topographic measurement
with an image size of 160 × 160 nm2 (512 by 512 pixels).
Figure 4(a) shows the full image; (b) and (c) are magnifications
of areas indicated by the frames. Figures 4(b) and (c) show the
centre of a screw dislocation. The step height at the left edge
is 314 pm. The image was recorded with a positive frequency
shift, i.e. with repulsive forces. Figure 4(d) shows the same
data as (a) with five-fold z-contrast. The small patches on
otherwise flat terraces are probably caused by local surface or
sub-surface charges.
Figure 5(a) shows the centre of a different screw
dislocation located 1.22 µm to the left of the screw dislocation
shown in figure 4(b). The profile in figure 5(b) shows that the
step height is initially only approximately 240 pm, followed
by another step with a height of only 80 pm. The unusual
step heights cannot be explained by the commonly accepted
crystallography of CaF2 (111). A reduced step height of
240 pm could possibly be explained by the long-range force
contributions explained in the text describing figure 3 (further
below, we find an experimental step height of only 275 pm for
a step imaged in the attractive mode; see figure 8). However,
a step with a height of 80 nm could not be explained by such
an effect. Other explanations, like a double tip effect, appear
unlikely because of the large lateral step distance of more than
40 nm.
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F J Giessibl and M Reichling
Figure 5. AFM image of a screw dislocation on a CaF2 (111)
surface. Imaging parameters: scanning speed 400 nm s−1 , image
size 150 × 150 nm2 , A = 1.25 nm, f = −0.74 Hz,
γ = −3.5 fN m0.5 .
A possibly obvious explanation appears to contradict
Tasker’s theorem [7]: a region with a width of 40 nm could
be stripped of the F− ions. In equilibrium, a (111) face
exposing Ca2+ ions is forbidden. On the other hand, cleaving
a crystal is not an equilibrium process, and our crystal was not
annealed after cleaving (figure 5(a) was taken about 30 min
after cleaving). Furthermore, the large strain fields in the
vicinity of a screw dislocation may help to violate the chargeneutrality principle. The large number of adsorbents on the
40 nm wide odd-stacked terrace points to a highly reactive
surface region, possibly caused by a large surface charge
density. We note that Ca2+ terminated surfaces of CaF2 (111)
have not been reported in the literature so far to our knowledge,
but neither have atomic images of screw dislocations. So far,
we have seen this unusual step height in one out of two screw
dislocation centres (incidentally, the second screw dislocation
centre shown in figure 4 was recorded five days after cleaving),
and further studies to elucidate the dislocation morphology of
CaF2 are planned.
Figure 6(a) shows a large scan showing approximately
12 000 atoms. There is a double triple-layer step in the right
bottom corner, but otherwise the terrace is flat. The µ-shaped
structure is possibly caused by surface charges. Repp et al [28]
have recently shown that charged Au-adatoms on NaCl can
cause a significant relaxation of the underlying NaCl lattice
that remains stable even when electrons tunnel through this
adatom. Discharging of this adatom and thus switching its
state is only possible when a voltage pulse with sufficient
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pulse height is applied. From Repp et al’s experiment, it is
conceivable that charged in-plane surface atoms on insulators
may also be stabilized by lattice distortions that could cause
slight deviations of an otherwise flat sample.
The magnified view in figure 6(b) shows a structural
defect, and figure 6(c) images the same area after 34 min. The
µ-shaped structure has disappeared in 6(c), but the magnified
view in 6(d) shows that the structural defect is still there. The
fast scan (0.25 lines s−1 , starting at the bottom in (a) and at the
top in (c)) was horizontal, the slow scan vertical. Thus, the step
at the right bottom in figure 6(c) was imaged 68 min after the
step in figure 6(a). Because of thermal drift, the steps appear
shifted in (a) and (c), and the magnified view of the structural
defect also shifts between figures 6(b) and (d).
Figure 7(a) is a high-resolution image taken in the attractive mode which, according to previous calculations [8, 12],
is produced by a positively terminated tip. The orientation
of the maxima, minima and saddle points in the image shows
that the sample is oriented as indicated in figures 1(b) and (c).
The maxima are attributed to the surface F− -layer, the minima
to the Ca2+ -layer that is 79 pm lower, and the saddle points
to the second F− -layer that is 158 pm lower than the surface
layer (see figure 1(b)). Figure 7(b) shows the same area imaged in a repulsive mode. As predicted by Foster et al [12],
the contrast changes: while the absolute minima in figure 7(a)
are adjacent to the right of the absolute maxima, they are left
of the absolute maxima in the repulsive data shown in figure 7(b). The magnitude of the γ -contrast in figure 7(a) of
±1.4 fN m0.5 agrees very well with the calculated value of
≈±1 fN m0.5 (Foster et al find a contrast of up to 8 Hz using
a cantilever with k = 6 N m−1 , f 0 = 84 kHz and A = 23 nm;
see pp 327 and 333 in [12]). The absolute value for γ according to theory is −38.6 fN m0.5 (see p 328 in [12]), while
we find almost twice that value in figure 7(b). This deviation is most likely due to the fairly large radius of our tip. In
large-amplitude FM-AFM, experimental values for γ range
from approximately −250 fN m0.5 [6] to −85 fN m0.5 [5]. In
the small-amplitude experiments presented here, we observed
atomic resolution in the attractive mode in a range from −100
to −25 fN m0.5 . The agreement of the parameter range for γ
in large- and small-amplitude regimes, where the basic imaging parameters k, A, f0 and f differ by orders of magnitude,
underlines the validity of γ as a universal figure describing
tip–sample interaction in FM-AFM.
Figure 8(a) is a topographic image across a monostep with
a height of 315 pm recorded in the attractive mode. Due to the
lateral variation of the vdW force, the apparent step height
is only 275 pm, but atomic resolution is present on both the
higher and the lower terrace. The setpoint of the normalized
frequency shift was γ = −16 fN m0.5 in this image, but due
to finite feedback speed, the actual frequency shift has shown
small variations, as shown in figure 8(b). The line analysis in
figure 8(c) of the error signal along the black line in figure 8(b)
shows that the F− ions are shifted by approximately 223 pm
to the right, as expected from figure 1(c).
Figure 9 demonstrates that the front atom of the tip is not a
metal atom from the original tip but consists of an ion or ionic
cluster picked up from the surface. The top section shows
contrast as expected from a negative tip termination, and the
bottom shows inverted contrast as if the charge of the front
Investigating atomic details of the CaF2 (111) surface with a qPlus sensor
Figure 6. High-resolution topographic images of CaF2 (111). Imaging parameters: scanning speed 20 nm s−1 (fast scan horizontal, slow
scan vertical starting at the bottom in (a) and at the top in (c)), image size 40 × 40 nm2 ((a), (c)), 6.1 × 6.1 nm2 ((b), (d)), A = 625 pm,
f = +3.66 Hz, γ = 6.15 fN m0.5 . The acquisition time for (a) and (c) is 34 min; (c) was taken right after (a).
Figure 7. High-resolution constant height images of CaF2 (111) in (a) the attractive and (b) the repulsive mode. Imaging parameters: image
size 1.36 × 1.36 nm2 , (a) scanning speed 4 nm s−1 , A = 1.25 nm, f = −13.3 ± 0.3 Hz, γ = −63 ± 1.4 fN m0.5 , (b) scanning speed
16 nm s−1 , A = 625 pm, f = +4.4 ± 0.3 Hz, γ = +7.4 ± 1 fN m0.5 .
atom is inverted. Such a contrast change could be caused by
a CaF2 tip cluster that is shifted or flipped during the scan,
exposing a F− ion initially and a Ca2+ tip ion in the lower
section of the image.
5. Conclusion and summary
In conclusion, we have shown that the use of stiff cantilevers
such as the qPlus sensor operated with small amplitudes allows
the imaging of steps on CaF2 (111) with atomic resolution both
in attractive and repulsive mode. Large terraces with up to
12 000 atoms have been imaged. The theoretical prediction
about a shift in contrast when switching from attractive to
repulsive imaging [8, 12] has been verified. The centres of
screw dislocations have been imaged for the first time, and we
found that small sample areas can exist that are not terminated
by complete triple layers. Further improvements are expected
when using qPlus sensors with sharp tips [14], because that
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F J Giessibl and M Reichling
would allow the further attenuation of the disturbing longrange contributions.
Acknowledgments
This work is supported by the Bundesministerium für Bildung
und Forschung (project EKM13N6918). We thank Ch Schiller
for the preparation of the qPlus sensor used in this study and
J Mannhart for support. MR gratefully acknowledges support
from the Deutsche Forschungsgemeinschaft.
References
Figure 8. High-resolution topographic image (a) and frequency
shift image (b) of CaF2 (111) across a single step. The measured
step height is 275 pm, 13% smaller than the expected step height of
315 pm (see the text). Imaging parameters: scanning speed
8.7 nm s−1 , image size 8.6 × 4.5 nm2 , A = 625 pm,
f = −9.5 Hz, γ = −16 ± 0.9 fN m0.5 . (c) Contour line along the
black trace indicated in (b). A filtered version of (b) was used to
produce the contour.
Figure 9. Topographic image of CaF2 (111) during which a tip
change occurs. Imaging parameters: scanning speed 4.8 nm s−1 ,
image size 2.9 × 2.9 nm2 , A = 625 pm, f = −8.8 Hz,
γ = −14.8 fN m0.5 .
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